Webbed space

In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.

Web
Let $$X$$ be a Hausdorff locally convex topological vector space. A  is a stratified collection of disks satisfying the following absorbency and convergence requirements.
 * 1) Stratum 1: The first stratum must consist of a sequence $$D_{1}, D_{2}, D_{3}, \ldots$$ of disks in $$X$$ such that their union $$\bigcup_{i \in \N} D_i$$ absorbs $$X.$$
 * 2) Stratum 2: For each disk $$D_i$$ in the first stratum, there must exists a sequence $$D_{i1}, D_{i2}, D_{i3}, \ldots$$ of disks in $$X$$ such that for every $$D_i$$: $$D_{ij} \subseteq \left(\tfrac{1}{2}\right) D_i \quad \text{ for every } j$$ and $$\cup_{j \in \N} D_{ij}$$ absorbs $$D_i.$$ The sets $$\left(D_{ij}\right)_{i,j \in \N}$$ will form the second stratum.
 * 3) Stratum 3: To each disk $$D_{ij}$$ in the second stratum, assign another sequence $$D_{ij1}, D_{ij2}, D_{ij3}, \ldots$$ of disks in $$X$$ satisfying analogously defined properties; explicitly, this means that for every $$D_{i,j}$$: $$D_{ijk} \subseteq \left(\tfrac{1}{2}\right) D_{ij} \quad \text{ for every } k$$ and $$\cup_{k \in \N} D_{ijk}$$ absorbs $$D_{ij}.$$ The sets $$\left(D_{ijk}\right)_{i,j,k \in \N}$$ form the third stratum.

Continue this process to define strata $$4, 5, \ldots.$$ That is, use induction to define stratum $$n + 1$$ in terms of stratum $$n.$$

A  is a sequence of disks, with the first disk being selected from the first stratum, say $$D_i,$$ and the second being selected from the sequence that was associated with $$D_i,$$ and so on. We also require that if a sequence of vectors $$(x_n)$$ is selected from a strand (with $$x_1$$ belonging to the first disk in the strand, $$x_2$$ belonging to the second, and so on) then the series $$\sum_{n = 1}^{\infty} x_n$$ converges.

A Hausdorff locally convex topological vector space on which a web can be defined is called a .

Examples and sufficient conditions
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All of the following spaces are webbed:
 * Fréchet spaces.
 * Projective limits and inductive limits of sequences of webbed spaces.
 * A sequentially closed vector subspace of a webbed space.
 * Countable products of webbed spaces.
 * A Hausdorff quotient of a webbed space.
 * The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.
 * The bornologification of a webbed space.
 * The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed.
 * If $$X$$ is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the continuous dual space of $$X$$ with the strong topology is webbed.
 * So in particular, the strong duals of locally convex metrizable spaces are webbed.
 * If $$X$$ is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.

Theorems
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If the spaces are not locally convex, then there is a notion of web where the requirement of being a disk is replaced by the requirement of being balanced. For such a notion of web we have the following results:

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